A classification of permutation binomials of the form x^i+ax over 𝔽_2^n for dimensions up to 8

Permutation polynomials with few terms (especially permutation binomials) attract many people due to their simple algebraic structure. Despite the great interests in the study of permutation binomials, a complete characterization of permutation binomials is still unknown. In this paper, we give a classification of permutation binomials of the form x^i+ax over 𝔽_2^n, where n≤ 8 by characterizing three new classes of permutation binomials. In particular one of them has relatively large index q^2+q+1/3 over 𝔽_q^3.